05/05/2009 Fall 2008 Advanced Topics (EENG 4010) Control Systems Design (EENG 5310)

What is a Control System? System- a combination of components that act together and perform a certain objective Control System- a system in which the objective is to control a process or a device or environment Process- a progressively continuing operations/development marked by a series of gradual changes that succeed one another in a relatively fixed way and lead towards a particular result or end.

05/05/2009 Control Theory Branch of systems theory (study of interactions and behavior of a complex assemblage) Open Loop Control System Control System Manipulated Variable(s) Control Variable(s) Manipulated Variable(s) Closed Loop Control System Control System Control Variable(s) Feedback function

Classification of Systems Classes of Systems Lumped Parameter Distributed Parameter (Partial Differential Equations, Transmission line example) Deterministic Stochastic Continuous Time Discrete Time Linear Nonlinear Constant Coefficient Time Varying Homogeneous (No External Input; system behavior depends on initial conditions) Non-homogeneous

Example Control Systems Mechanical and Electo-mechanical (e.g. Turntable) Control Systems Thermal (e.g. Temperature) Control System Pneumatic Control System Fluid (Hydraulic) Control Systems Complex Control Systems Industrial Controllers On-off Controllers Proportional Controllers Integral Controllers Proportional-plus-Integral Controllers Proportional-plus-Derivative Controllers Proportional-plus-Integral-plus-Derivative Controllers

Mathematical Background Why needed? (A system with differentials, integrals etc.) Complex variables (Cauchy-Reimann Conditions, Euler Theorem) Laplace Transformation Definition Standard Transforms Inverse Laplace Transforms Z-Transforms

Laplace Transform Definition Condition for Existence Laplace Transforms of exponential, step, ramp, sinusoidal, pulse, and impulse functions Translation of and multiplication by Effect of Change of time scale Real and complex differentiations, initial and final value theorems, real integration, product theorem Inverse Laplace Transform

Inverse Laplace Transform 05/05/2009 Inverse Laplace Transform Definition Formula is seldom or never used; instead, Heaviside partial fraction expansion is used. Illustration with a problem: Initial conditions: y(0) = 1, y’(0) = 0, and r(t) = 1, t >= 0. Find the steady state response Multiple pole case with Use the ideas to find and

Applications Spring-mass-damper- Coulomb and viscous damper cases 05/05/2009 Applications Spring-mass-damper- Coulomb and viscous damper cases RLC circuit, and concept of analogous variables Solution of spring-mass-damper (viscous case) DC motor- Field current and armature current controlled cases Block diagrams of the above DC-motor problems Feedback System Transfer functions and Signal flow graphs

Block Diagram Reduction 05/05/2009 Block Diagram Reduction Combining blocks in a cascade Moving a summing point ahead of a block Moving summing point behind a block Moving splitting point ahead of a block Moving splitting point behind a block Elimination of a feedback loop H2 Y(s) - + R(s) G1 G2 G3 G4 + - + H1 H3

Signal Flow Graphs Mason’s Gain Formula Solve these two equations and generalize to get Mason’s Gain Formula r1 x1 a21 a12 x2 r2 a22 H2 H3 G1 G2 G3 G4 R(s) Y(s) G6 G7 G8 G5 Find Y(s)/R(s) using the formula H8 H7

Control System Stability: Routh-Hurwitz Criterion Why poles need to be in Right Hand Plane Necessary condition involving Characteristic Equation (Polynomial) Coefficients Proof that the above condition is not sufficient Ex: s3+s2+2s+8. Routh-Hurwitz Criterion- Necessary & Sufficient

Routh-Hurwitz Criterion: Some Typical Problems 2nd and 3rd order systems q(s)=s5+2s4+2s3+4s2+11s+10 (first element of a row 0; other elements are not) q(s)=s4+s3+s2+s+K (Similar to above case) q(s)=s3+2s2+4s+K (for k = 8, first element of a row 0; so are other elements of the row) q(s)= s5+s4+2s3+2s2+s+1: Repeated roots on imaginary axis; Marginally stable case

Root-Locus Method: What and Why? 05/05/2009 Root-Locus Method: What and Why? Plotting the trajectories of the poles of a closed loop control system with free parameter variations Useful in the design for stability with out sacrificing much on performance Closed Loop Transfer Function Let Open Loop Gain Roots of the closed loop characteristic equation depend on K. R(s) Y(s) + - G(s) H(s)

Relationship between closed loop poles and open loop gain When K=0, closed loop poles match open loop poles When closed loop poles match open loop zeros. Hence we can say, the closed loop poles start at open loop poles and approach closed loop zeros as K increases and thus form trajectories.

Mathematical Preliminaries of Root Locus Method 05/05/2009 Mathematical Preliminaries of Root Locus Method Complex numbers can be expressed as (absolute value, angle) pairs. Now, The loci of closed loop poles can be determined using the above constraints (particularly, the angle constraint) on G(s)H(s). |s| s=s+jw w q s s=|s|.ejq =|s| q |s+s1| w s=s+jw s f -s1=-s1-jw1 s+s1=|s+s1|ejf

Root Locus Method- Step1 thru 3 of a 7-Step Procedure Step-1: Locate poles and zeros of G(s)H(s). Step 2: Determine Root Locus on the real-axis using angle constraint. Value of K at any particular test point s can be calculated using the magnitude constraint. Step 3: Find asymptotes by using angle constraint in . Find asymptote centroid . This formula may be obtained by setting Illustrative Problem: -1 -2

Root Locus Method- Step 4 Step 4: Determine breakaway points (points where two or more loci coincide giving multiple roots and then deviate). Now, from the characteristic equation where , we get, at a multiple pole s1, , because at s1, . Thus we get at s1, Since at s = s1. Thus, we get break points by setting dK/ds=0. In the example, we get s = -0.4226 or -1.5774 (invalid).

Root Locus Method- Step 5 Step 5: Determine the points (if any) where the root loci cross the imaginary axis using Roth-Hurwitz Stability Criterion. Illustration with the Example Problem Characteristic equation for the problem:s3+3s2+2s+K From the array, we know that the system is marginally stable at K=6. Now, we can get the value of w (imaginary axis crossing) either by solving the second row 3s2+6 =0 or the original equation with s=jw. S3 1 2 S2 3 K S1 (6-K)/3 S0

Root Locus Method- Step 6 and 7 Step 6: Determine angles of departure at complex poles and arrival at complex zeros using angle criterion. Step 7: Choose a test point in the broad neighborhood of imaginary axis and origin and check whether sum of the angles is an odd multiple of +180 or -180. If it does not satisfy, select another one. Continue the process till sufficient number of test points satisfying angle condition are located. Draw the root loci using information from steps 1-5.

Root Locus approach to Control System Design 05/05/2009 Root Locus approach to Control System Design Effect of Addition of Poles to Open Loop Function: Pulls the root locus right; lowers system’s stability and slows down the settling of response. Effect of Addition of Zeros to Open Loop Function: Pulls the Root Locus to Left; improves system stability and speeds up the settling of response s jw x s jw x s jw x x x x jw jw jw o x x x x o x x x x o x s s s

Performance Criteria Used In Design We consider 2nd order systems here, because higher order systems with 2 dominant poles can be approximated to 2nd order systems e.g. when For 2nd order system For unit step input Where . Two types of performance criteria (Transient and Steady State) Stability is a validity criterion (Non-negotiable).

Transient Performance Criteria overshoot ess y(t) 1.0 Rise Time TR= Time to reach Value 1.0 Rise Time Tr1= Time from 0.1 to 0.9 Empirical Formula is for Settling time (Time to settle to within 98% of 1.0)=4/xwn Peak Time Percentage Overshoot = TP t TR TS wnTr1 2.0 x 0.6

Series Compensators for Improved Design RC OP-Amp Circuit for phase lead (or lag) compensator Lead Compensator for Improved Transient Response; Example: Required to reduce rise time to half keeping x = 0.5. Lag Compensator for Improved steady-state performance. Example:

Frequency Response Analysis Response to x(t) = X sin(wt) G(s) = K/(Ts+1) and G(s)=(s+1/T1)/(s+1/T2) cases Frequency response graphs- Bode, and Nyquist plots of Resonant frequency and peak value Nichols Chart Nyquist Stability Criterion

Control System Design Using Frequency Response Analysis Lead Compensation Lag Compensation Lag-Lead Compensation

State Space Analysis State-Space Representation of a Generic Transfer Function in Canonical Forms: Controllable Canonical Form Observable Canonical Form Diagonal Canonical Form Jordan Canonical Form Eigenvalue Analysis

Solution of State Equations Solution of Homogeneous Equations Interpretation of Show that the state transition matrix is given by Properties of Solution of Nonhomogeneous Equations Cayley-Hamilton Theorem

Controllability and Observability Definitions of Controllable and Observable Systems Controllabililty and Obervability Conditions Principle of Duality

Control System Design in State Space Necessary and Sufficient Condition for Arbitrary Pole Placement Determination of Feedback Gain Matrix by Ackerman’s formula Design of Servo Systems

Introduction to Sampled Data Control Systems Z-transform and Inverse Z-transform Properties of Z-Transform and Comparison with the Corresponding Laplace Transform Properties Transfer Functions of Discrete Data Systems

Analysis of Sampled Data Systems Input and Output Response of Sampled Data Systems Differences in the Transient Characteristics of Continuous Data Systems and Corresponding Discrete (Sampled) Data Systems Root Locus Analysis of Sampled Data Systems